Black-Litterman made an assumption about $\Sigma_\pi$, which is proportional to the covariance of the returns $\Sigma$ with a scalar $\tau$. There are many different ways to choose $\tau$ in the literatures. We follow the class's suggestion to use $\tau = \frac{1}{T} = 0.02$, based on MLE estimate, where T is the length of the sample period (5 years). The B-L model takes a projection matrix $P$ and investor specified view vector $Q$ to compute the combined posterior return vector $\mu^{BL}$ and $\Sigma^{BL}$. In our case, we predicted the absolute (not relative) the returns for each portfolio. Hence, our vector $Q$ is the column vector of the predicted returns for the current period. $P$ is then an identity matrix according to the definition. We follow the B-L rules to estimate the posterior $\mu^{BL}$ and $\Sigma^{BL}$ by:

\begin{eqnarray*}
\mu^{BL} &=& [(\tau \Sigma)^{-1} + P^T \Omega^{-1} P]^{-1}[(\tau \Sigma)^{-1}\pi + P^T \Omega^{-1}Q] \\
\Sigma^{BL} &=& \Sigma + [(\tau \Sigma)^{-1} + P^T \Omega^{-1} P]^{-1}]
\end{eqnarray*}

In order to conduct B-L estimation, we also need to specify our confidence about the views. This is represented by $\Omega$, which is a diagonal covariance matrix of error terms from the expressed view representing the uncertainty in each view. The off-diagonal elements of $\Omega$ are 0's because the model assumes that the views are independent of one another. We calculate $\Omega$ by the following form:

\begin{eqnarray*}
\Omega &=& \left[ \begin{array} {ccc}
(p_1\Sigma p'_1) * \tau & 0 & 0 \\
0	& ... & 0 \\
0 	& ... & (p_6\Sigma p'_6) * \tau \\
\end{array} \right]
\end{eqnarray*}

in which  $p_1 = p_2 ... = p_6 = 1$ since each view is an absolute view in our models. E is the covariance matrix built for each portfolio.

After we compute $\mu^{BL}$ and $\Sigma^{BL}$, we use them to replace the plug-in estimate $\hat{\mu}$ and $\hat{\Sigma}$ as inputs to the M-V optimizer. We use the weights corresponding to the global MVP in the Mean-Variance efficient frontier as our asset allocation weights \footnote{We also tried to use the reverse optimization equation to induct weight: $\omega = (\lambda \Sigma)^{-1}\mu$, as shown in \cite{idzorek2005}. But, it leads to unreasonable results.}.  